We consider a fractional porous medium equation that extends the classical porous medium and fractional heat equations. The flow is studied in the space of periodic probability measures endowed with a non-local transportation distance constructed in the spirit of the Benamou-Brenier formula. For initial periodic probability measures, we show the existence of absolutely continuous curves that are generalized minimizing movements associated to Rényi entropy. For that, we need to obtain entropy and distance properties and to develop a subdifferential calculus in our setting.
We prove that a general condition introduced by Colombo and Gobbino to study limits of curves of maximal slope allows also to characterize minimizing movements along a sequence of functionals as curves of maximal slope of a limit functional.
Curves of maximal slope are a reference gradient-evolution notion in metric spaces and arise as variational formulation of a vast class of nonlinear diffusion equations. Existence theories for curves of maximal slope are often based on minimizing-movements schemes, most notably on the Euler scheme. We present here an alternative minimizing-movements approach, yielding more regular discretizations, serving as a-posteriori convergence estimator, and allowing for a simple convergence proof.
We study an $L^{2}$-type gradient flow of an immersed elastic curve in $\mathbb{R}^{2}$ whose endpoints repel each other via a Coulomb potential. By De Giorgi's minimizing movements scheme we prove long-time existence of the flow. The work is complemented by several numerical experiments.
We study the mean curvature motion of a droplet flowing by mean curvature on a horizontal hyperplane with a possibly nonconstant prescribed contact angle. Using the minimizing movements method we show the existence of a weak evolution, and its compatibility with a distributional solution. We also prove various comparison results.
We present new abstract results on the interrelation between the minimizing movement scheme for gradient flows along a sequence of Gamma-converging functionals and the gradient flow motion for the corresponding limit functional, in a general metric space. We are able to allow a relaxed form of minimization in each step of the scheme, and so we present new relaxation results too.
The gradient flow of the Canham-Helfrich functional is tackled via the Generalized Minimizing Movements approach. We prove the existence of solutions in Wasserstein spaces of varifolds, as well as upper and lower diameter bounds. In the more regular setting of multiply covered $C^{1,1}$ surfaces, we provide a Li-Yau-type estimate for the Canham-Helfrich energy and prove the conservation of multiplicity along the evolution.
The purpose of this paper is to introduce a Minimizing Movement approach to a class of scalar reaction-diffusion equations, which is built on their gradient-flow-like structure in the space of finite nonnegative Radon measures, endowed with the recently introduced Hellinger-Kantorovich distance. Moreover, a superdifferentiability property of the Hellinger-Kantorovich distance, which will play an important role in this context, is established in the general setting of a separable Hilbert space.
We prove the existence of a weak global in time mean curvature flow of a bounded partition of space using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We also prove some consistency results for a minimizing movement solution with smooth and viscosity solutions when the evolution starts from a partition made by a union of bounded sets at a positive distance. In addition, the motion starting from the union of convex sets at a positive distance agrees with the classical mean curvature flow and is stable with respect to the Hausdo…
We recast the Calabi flow in DeGiorgi's language of minimizing movements. We establish the long time existence of minimizing movements for K-energy with arbitrary initial condition. Furthermore we establish some a priori regularity of these solutions, and that sufficiently regular minimizing movements are smooth solutions to Calabi flow.
Under suitable assumptions on the family of anisotropies, we prove the existence of a weak global $\frac{1}{n+1}$-Hölder continuous in time mean curvature flow with mobilities of a bounded anisotropic partition in any dimension using the method of minimizing movements. The result is extended to the case when suitable driving forces are present. We improve the Hölder exponent to $\frac12$ in the case of partitions with the same anisotropy and the same mobility and provide a weak comparision result in this setting for a weak anisotropic mean curvature flow of a partition and an anisotropic mean …
We consider the thresholding scheme and explore its connection to De Giorgi's ideas on gradient flows in metric spaces; here applied to mean curvature flow as the steepest descent of the interfacial area. The basis of our analysis is the observation by Esedoglu and the second author that thresholding can be interpreted as a minimizing movements scheme for an energy that approximates the interfacial area. De Giorgi's framework provides an optimal energy dissipation relation for the scheme in which we pass to the limit to derive a dissipation-based weak formulation of mean curvature flow. Althou…
We consider the well-known minimizing-movement approach to the definition of a solution of gradient-flow type equations by means of an implicit Euler scheme depending on an energy and a dissipation term. We perturb the energy by considering a ($Γ$-converging) sequence and the dissipation by varying multiplicative terms. The scheme depends on two small parameters $\varepsilon$ and $τ$, governing energy and time scales, respectively. We characterize the extreme cases when $\varepsilon/τ$ and $τ/\varepsilon$ converges to $0$ sufficiently fast, and exhibit a sufficient condition that guarantees th…
We prove existence of variational solutions for a class of doubly nonlinear nonlocal evolution equations whose prototype is the double phase equation \begin{align*} \partial_t u^m &+ \text{P.V.}\int_{\mathbb{R}^N} \frac{|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{N+ps}}\\&+a(x,y)\frac{|u(x,t)-u(y,t)|^{q-2}(u(x,t)-u(y,t))}{|x-y|^{N+qr}} \,dy = 0,\,m>0,\,p>1,\,s,r\in (0,1). \end{align*} We make use of the approach of minimizing movements pioneered by DeGiorgi and Ambrosio and refined by Bögelein, Duzaar, Marcellini, and co-authors to study nonlinear parabolic equations with non-standard growth.
We prove the existence of nonnegative weak solutions to a class of second and fourth order nonautonomous nonlinear evolution equations with an explicitly time-dependent mobility function posed on the whole space $\mathbb{R}^d$, for arbitrary $d\ge 1$. Exploiting a very formal gradient flow structure, the cornerstone of our proof is a modified version of the classical minimizing movement scheme for gradient flows. The mobility function is required to satisfy - at each time point separately - the conditions by which one can define a modified Wasserstein distance on the space of probability densi…
Motivated by models of fracture mechanics, this paper is devoted to the analysis of unilateral gradient flows of the Ambrosio-Tortorelli functional, where unilaterality comes from an irreversibility constraint on the fracture density. In the spirit of gradient flows in metric spaces, such evolutions are defined in terms of curves of maximal unilateral slope, and are constructed by means of implicit Euler schemes. An asymptotic analysis in the Mumford-Shah regime is also carried out. It shows the convergence towards a generalized heat equation outside a time increasing crack set.
We perform a convergence analysis of a discrete-in-time minimization scheme approximating a finite dimensional singularly perturbed gradient flow. We allow for different scalings between the viscosity parameter $\varepsilon$ and the time scale $τ$. When the ratio $\frac{\varepsilon}τ$ diverges, we rigorously prove the convergence of this scheme to a (discontinuous) Balanced Viscosity solution of the quasistatic evolution problem obtained as formal limit, when $\varepsilon\to 0$, of the gradient flow. We also characterize the limit evolution corresponding to an asymptotically finite ratio betwe…
Simulation of multiphase porous media flows with minimizing movement and finite volume schemes
Inria, Univ. Lille, CNRS, UMR 8524 - Laboratoire Paul Painlevé（法国国家信息与自动化研究所，里尔大学，法国国家科学研究中心）；Inria, Project team MOKAPLAN and Mathematics department, Université de Liège（法国国家信息与自动化研究所，列日大学）；Department of Mathematics and Statistics, McGill University（麦吉尔大学数学与统计系）；IECL Université de Lorraine（洛林大学计算数学与复杂系统研究所）；GFM Universidade de Lisboa（里斯本大学）
In this paper we investigate the interaction of fluid flow with a thin porous elastic layer. We consider two fluid-filled bulk domains which are separated by a thin periodically perforated layer consisting of a fluid and an elastic solid part. Thickness and periodicity of the layer are of order $ε$, where $ε$ is small compared to the size of the bulk domains. The fluid flow is described by an instationary Stokes equation and the solid via linear elasticity. The main contribution of this paper is the rigorous homogenization of the porous structure in the layer and the reduction of the layer to …
Global existence and boundedness for a degenerate chemotaxis system with indirect signal production via minimizing movement schemes
Osaka Metropolitan University（大阪公立大学）；Institut de Mathématiques de Toulouse (IMT) UMR5219, Université Toulouse III - Paul Sabatier（图卢兹第三大学-保罗·萨巴蒂尔大学）；Université Savoie Mont Blanc, CNRS（萨瓦-蒙布朗大学）
Recently solutions to the aggregation equation on compact Riemannian Manifolds have been studied with different techniques. This work demonstrates the small time existence of measure-valued solutions for suitably regular intrinsic potentials. The main tool is the use of the minimizing movement scheme which together with the optimality conditions yield a finite speed of propagation. The main technical difficulty is non-differentiability of the potential in the cut locus which is resolved via the propagation properties of geodesic interpolations of the minimizing movement scheme and passes to th…
De Giorgi varifold solutions to Mean Curvature Flow: a minimizing movements approach
Institut für Mathematik & Interdisziplinäres Zentrum für Wissenschaftliches Rechnen, Universität Heidelberg（海德堡大学数学与跨学科科学计算中心）；Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università degli Studi di Parma（帕尔马大学数学、物理与信息科学系）
Convergence of a minimizing movement scheme for contact-angle mean curvature flow in a smooth bounded domain
Université Claude Bernard Lyon 1, CNRS, Centrale Lyon, INSA Lyon, Université Jean Monnet, ICJ UMR5208（里昂第一大学、法国国家科学研究中心、中央里昂学院、里昂国立应用科学学院、让·莫内大学、ICJ联合研究实验室）；Department of Mathematics, University of Maryland-College Park（马里兰大学帕克分校数学系）
We consider the relationship between Hele-Shaw evolution with drift, the porous medium equation with superharmonic drift, and a congested crowd motion model originally proposed by [MRS]- [MRSV]. We first use viscosity solutions to show that the porous medium equation solutions converge to the Hele- Shaw solution as the exponent tends to infinity. Next, using of the gradient flow structure of both the porous medium equation and the crowd motion model, we prove that the porous medium equation solutions also converge to the congested crowd motion. Combining these results lets us deduce that in th…
We consider the Cauchy problem for the gradient flow \begin{equation} \label{eq:81} \tag{$\star$} u'(t)=-\nablaϕ(u(t)),\quad t\ge 0;\quad u(0)=u_0, \end{equation} generated by a continuously differentiable function $ϕ:\mathbb H \to \mathbb R$ in a Hilbert space $\mathbb H$ and study the reverse approximation of solutions to ($\star$) by the De Giorgi Minimizing Movement approach. We prove that if $\mathbb H$ has finite dimension and $ϕ$ is quadratically bounded from below (in particular if $ϕ$ is Lipschitz) then for every solution $u$ to ($\star$) (which may have an infinite number of solution…
We study the asymptotic behavior of a discrete-in-time minimizing movement scheme for square lattice interfaces when both the lattice spacing and the time step vanish. The motion is assumed to be driven by minimization of a weighted random perimeter functional with an additional deterministic dissipation term. We consider rectangular initial sets and lower order random perturbations of the perimeter functional. In case of stationary, $α$-mixing perturbations we prove a stochastic homogenization result for the interface velocity. We also provide an example which indicates that stationary, ergod…
We prove existence of weak solutions to the obstacle problem for semilinear wave equations (including the fractional case) by using a suitable approximating scheme in the spirit of minimizing movements. This extends the results in [9], where the linear case was treated. In addition, we deduce some compactness properties of concentration sets (e.g. moving interfaces) when dealing with singular limits of certain nonlinear wave equations.
This paper deals with the limit cases for $s$-fractional heat flows in a cylindrical domain, with homogeneous Dirichlet boundary conditions, as $s\to 0^+$ and $s\to 1^-$\,. To this purpose, we describe the fractional heat flows as minimizing movements of the corresponding Gagliardo seminorms, with respect to the $L^2$ metric. First, we provide an abstract stability result for minimizing movements in Hilbert spaces, with respect to a sequence of $Γ$-converging uniformly $λ$-convex energy functionals. Then, we provide the $Γ$-convergence analysis of the $s$-Gagliardo seminorms as $s\to 0^+$ and …
In fluid flow simulation, the multi-continuum model is a useful strategy. When the heterogeneity and contrast of coefficients are high, the system becomes multiscale, and some kinds of reduced-order methods are demanded. Combining these techniques with nonlinearity, we will consider in this paper a dual-continuum model which is generalized as a multi-continuum model for a coupled system of nonlinear Richards equations as unsaturated flows, in complex heterogeneous fractured porous media; and we will solve it by a novel multiscale approach utilizing the constraint energy minimizing generalized …
